HalfSpace
Details
- HalfSpace can be used as a geometric region and a graphics primitive.
- HalfSpace corresponds to half-line or half-infinite interval in , a half-plane in , etc.
- HalfSpace represents the set or .
- HalfSpace can be used in Graphics and Graphics3D .
- HalfSpace will be clipped by PlotRange when rendering.
- Graphics rendering is affected by directives such as FaceForm , EdgeForm , Opacity , and color.
- Graphics3D rendering is affected by directives such as Opacity and color.
Examples
open all close allBasic Examples (3)
A HalfSpace in 2D:
And in 3D:
Different styles applied to a half-space region:
Determine if points belong to a given half-space region:
Scope (15)
Graphics (5)
Specification (2)
A half-space in 2D defined by a normal vector and a point:
The same half-space defined by a normal vector and a constant:
Define a half-space in 3D using a normal vector and a point:
Define the same half-space using a normal vector and a constant:
Half-spaces varying in direction of the normal:
Styling (2)
Coordinates (1)
Points and vectors can be Dynamic :
Regions (10)
Embedding dimension is the dimension of the coordinates:
Geometric dimension is the dimension of the region itself:
Point membership test:
Get the conditions for membership:
A half-space has infinite measure and undefined centroid:
Distance from a point:
Signed distance from a point:
Nearest point in the region:
Nearest points:
A half-space is unbounded:
Find the region range:
In the axis-aligned case:
Integrate over a half-space:
Optimize over a half-space:
Solve equations over a half-space:
Applications (5)
Visualize 2D half-planes:
The upper half-plane:
The lower half-plane:
The left half-plane:
The right half-plane:
Visualize 3D half-spaces:
The upper half-space:
The lower half-space:
The left half-space:
The right half-space:
The front half-space:
The back half-space:
Partition space in a BubbleChart :
Combine the graphics:
Any convex polygon in 2D can be represented as an intersection of half-spaces:
Any convex polyhedron in 3D can be represented as an intersection of half-spaces:
Properties & Relations (7)
ClipPlanes , for a given , results in a graphic that does not render anything within the :
HalfSpace is a special case of ConicHullRegion :
HalfSpace is a special case of AffineHalfSpace :
HalfLine is a special case of HalfSpace :
HalfPlane is a special case of HalfSpace :
ImplicitRegion can represent any HalfSpace in :
In :
In :
ParametricRegion can represent any HalfSpace in :
In :
In :
Neat Examples (1)
A collection of random half-spaces in :
In :
See Also
Related Guides
History
Text
Wolfram Research (2015), HalfSpace, Wolfram Language function, https://reference.wolfram.com/language/ref/HalfSpace.html.
CMS
Wolfram Language. 2015. "HalfSpace." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HalfSpace.html.
APA
Wolfram Language. (2015). HalfSpace. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HalfSpace.html
BibTeX
@misc{reference.wolfram_2025_halfspace, author="Wolfram Research", title="{HalfSpace}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/HalfSpace.html}", note=[Accessed: 04-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_halfspace, organization={Wolfram Research}, title={HalfSpace}, year={2015}, url={https://reference.wolfram.com/language/ref/HalfSpace.html}, note=[Accessed: 04-January-2026]}